Equivalent Length vs. Minor Pressure Head Loss in Pipe and Duct Components

Minor pressure and head loss in pipes vs. equivalent length in tubes and duct systems.

Minor Loss

The pressure drop or the minor loss in a component correlates to the dynamic pressure in the flow and can be expressed as

Δp_{minor_loss} = ξ p_d

= ξ ρ_f v² / 2 (1)

where

ξ = minor loss coefficient

p_d = dynamic pressure in fluid flow (Pa (N/m²), psf (lb/ft²))

Δp_{minor_loss} = minor pressure loss (Pa (N/m²), psf (lb/ft²))

ρ_f = density of fluid (kg/m³, slugs/ft³)

v = flow velocity (m/s, ft/s)

The minor loss can be expressed as head water column by dividing the dynamic pressure with the specific weight of water

Δh_{minor_loss,w}= (ξ ρ_fv²/ 2) / γ_w

= (ξ ρ_fv²/ 2) / (ρ_w g)

= ξ ρ_f v²/ (2 ρ_w g) (2)

where

Δh_{minor_loss,w} = minor head loss as water column (m H2O, ft H2O)

γ_w = ρ_w g = specific weight of water or reference fluid (9807 N/m³, 62.4 lb_f/ft³)

g = acceleration of gravity (9.81 m/s², 32.174 ft/s²)

1 psf = 0.00694 psi (lb/in²)

Note! - in the equation above the head is related to water as the reference fluid. Another reference fluid can be used - like Mercury Hg - by replacing the density of water with the density of the reference fluid - check Velocity Pressure Head.

If the flowing fluid has the same density as the reference fluid - typical for a water flow - eq. (2) can simplified to

Δh_{minor_loss}= ξ v²/ (2 g) (2b)

where

Δh_{minor_loss}= minor head loss (column of flowing fluid) (m fluid column, ft fluid column)

Minor Loss Coefficient

The minor loss coefficient - ξ - values ranges from 0 and upwards. For ξ = 0 the minor loss is zero and for ξ = 1 the minor loss is equal to the dynamic pressure or head. The minor loss coefficient can also be greater than 1 for some components.

The minor loss coefficient can be expressed by rearranging (1) to

ξ = 2 Δp_{minor_loss} / (ρ_f v²) (3)

The minor loss coefficient can alternatively be expressed by rearranging (2) to

ξ = 2 ρ_w g Δh_{minor_loss,w} / (ρ_f v²) (4)

The dynamic loss in components depends primarily on the geometrical construction of the component and the impact the construction has on the fluid flow due to change in velocity and cross flow fluid accelerations.

The fluid properties - in general expressed with the Reynolds number - also impacts the minor loss.

Minor loss information about components are given in dimensionless form based on experiments.

Equivalent Length

The dynamic minor loss in a component can be converted to an equivalent length of pipe or tube that would give the same major loss.

Major loss in a fluid flow can be expressed as

Δp_{major_loss} = λ (l / d_h) (ρ_f v² / 2) (5)

where

Δp_{major_loss} = major (friction) pressure loss in fluid flow (Pa (N/m²), psf (lb/ft²))

λ = Darcy-Weisbach friction coefficient

l = length of duct or pipe (m, ft)

v = velocity of fluid (m/s, ft/s)

d_h = hydraulic diameter (m, ft)

ρ_f = density of fluid (kg/m³, slugs/ft³)

If we want the minor loss to be equal to the major loss for a given equivalent length of pipe or duct - then

Δp_{minor_loss} = Δp_{major_loss, eq} (6)

or by combining (1) and (2)

ξ ρ_f v² / 2 = λ (l_eq / d_h) (ρ_f v² / 2) (6b)

where

Δp_{major_loss, eq} = equivalent major loss (Pa (N/m²), psf (lb/ft²))

l_eq = equivalent pipe length (m, ft)

(6b) can be reduced and rearranged to express equivalent length as

l_eq = ξ d_h / λ (7)

The total head loss in a pipe, tube or duct system, is the same as that produced in a straight pipe or duct whose length is equal to the pipes of the original systems - plus the sum of the equivalent lengths of all components in the system.

Example - Equivalent Length of Gate Valve

The equivalent length of a 50 mm gatevalve with loss coefficient 0.26 when 1/4 closed located in a steel pipe with friction coefficient 0.03 can be calculated with (7) as

l_eq = 0.26 (0.05 m) / 0.03

= 0.4 m